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# primitive root

Given any positive integer $n$, the group of units $U(\mathbb{Z}/n\mathbb{Z})$ of the ring $\mathbb{Z}/n\mathbb{Z}$ is a cyclic group iff $n$ is 4, $p^{m}$ or $2p^{m}$ for any odd positive prime $p$ and any non-negative integer $m$. A *primitive root* is a generator of this group of units when it is cyclic.

Equivalently, one can define the integer $r$ to be a primitive root modulo $n$, if the numbers $r^{0},\,r^{1},\,\ldots,\,r^{{n-2}}$ form a reduced residue system modulo $n$.

For example, 2 is a primitive root modulo 5, since
$1,\;2,\;2^{2}=4,\;2^{3}=8\equiv 3\;\;(\mathop{{\rm mod}}5)$
are all with 5 coprime positive integers less than 5.

The generalized Riemann hypothesis implies that every prime number $p$ has a primitive root below $70(\ln p)^{2}$.

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## Attached Articles

## Corrections

Notation, name by Koro ✓

suggestion by Mathprof ✓

minor things by CWoo ✓

riemann hypothesis by Mathprof ✓